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Properties

Label 16.0.14362567555...2129.2
Degree $16$
Signature $[0, 8]$
Discriminant $13^{6}\cdot 29^{14}$
Root discriminant $49.81$
Ramified primes $13, 29$
Class number $25$ (GRH)
Class group $[5, 5]$ (GRH)
Galois group $(C_2\times C_4).D_4$ (as 16T121)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![119525, -311475, 303248, -245902, 613596, -718469, 473915, -216549, 83734, -27415, 7377, -1854, 504, -109, 22, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 22*x^14 - 109*x^13 + 504*x^12 - 1854*x^11 + 7377*x^10 - 27415*x^9 + 83734*x^8 - 216549*x^7 + 473915*x^6 - 718469*x^5 + 613596*x^4 - 245902*x^3 + 303248*x^2 - 311475*x + 119525)
 
gp: K = bnfinit(x^16 - 6*x^15 + 22*x^14 - 109*x^13 + 504*x^12 - 1854*x^11 + 7377*x^10 - 27415*x^9 + 83734*x^8 - 216549*x^7 + 473915*x^6 - 718469*x^5 + 613596*x^4 - 245902*x^3 + 303248*x^2 - 311475*x + 119525, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 22 x^{14} - 109 x^{13} + 504 x^{12} - 1854 x^{11} + 7377 x^{10} - 27415 x^{9} + 83734 x^{8} - 216549 x^{7} + 473915 x^{6} - 718469 x^{5} + 613596 x^{4} - 245902 x^{3} + 303248 x^{2} - 311475 x + 119525 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1436256755503642932525262129=13^{6}\cdot 29^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.81$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $13, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{14} a^{14} + \frac{1}{7} a^{13} - \frac{1}{7} a^{12} - \frac{1}{7} a^{11} + \frac{1}{14} a^{10} + \frac{5}{14} a^{9} - \frac{1}{14} a^{8} - \frac{1}{14} a^{7} + \frac{2}{7} a^{6} + \frac{5}{14} a^{5} - \frac{1}{2} a^{3} - \frac{3}{14} a^{2} - \frac{1}{7} a$, $\frac{1}{1302737251140393179743205080883528370662710} a^{15} + \frac{1830381721000668755046589755090145194911}{93052660795742369981657505777394883618765} a^{14} - \frac{94398368670424287711816627622510587300364}{651368625570196589871602540441764185331355} a^{13} - \frac{226939892964562870677659059321576033976929}{1302737251140393179743205080883528370662710} a^{12} + \frac{149959194801302537850267665426356131926292}{651368625570196589871602540441764185331355} a^{11} - \frac{55436295734299750150754738155216765443872}{651368625570196589871602540441764185331355} a^{10} - \frac{165870568636260231740144149751623289892674}{651368625570196589871602540441764185331355} a^{9} + \frac{121200750569779114873507280834767250250425}{260547450228078635948641016176705674132542} a^{8} + \frac{13030146187203413764395481074743914557809}{1302737251140393179743205080883528370662710} a^{7} + \frac{299820247744349816183010382210227441381781}{1302737251140393179743205080883528370662710} a^{6} - \frac{108183812514758730554390731137532909017959}{260547450228078635948641016176705674132542} a^{5} + \frac{12126348675224681051816108131193801757619}{93052660795742369981657505777394883618765} a^{4} + \frac{122603666293553381001923638906635070786691}{1302737251140393179743205080883528370662710} a^{3} - \frac{274666719325990264995822982747193584424761}{651368625570196589871602540441764185331355} a^{2} - \frac{220623133764737651519952561009363885505277}{1302737251140393179743205080883528370662710} a - \frac{7963958814099408488770496975724885347435}{18610532159148473996331501155478976723753}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{5}\times C_{5}$, which has order $25$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1281839.59563 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_4).D_4$ (as 16T121):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $(C_2\times C_4).D_4$
Character table for $(C_2\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{29}) \), 4.0.24389.1, 4.4.317057.1, 4.0.10933.1, 8.8.37897978250873.1, 8.0.37897978250873.1, 8.0.100525141249.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.8.6.2$x^{8} + 39 x^{4} + 676$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
29Data not computed